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2012 Pólya Award Winners

August 6, 2012

The Mathematical Association of America has selected T.S. Michael (U.S. Naval Academy) and Leslie A. Cheteyan, Stewart Hengeveld, and Michael A. Jones (citation) as the 2012 winners of the George PÃ³lya Award. Full citations and biographical information for each winner are available below.

This award, established in 1976, is named after the renowned teacher and writer, and is given for articles of expository excellence published in the College Mathematics Journal. This is an award of $500. Up to two of these awards are given annually at the Summer Meeting of the Association.

Beginning with Victor Klee?s 1973 art gallery problem?to determine the maximum number of guards needed to protect an art gallery with a simple, closed polygonal floor plan?Michael offers a masterful and beautifully written survey of art-gallery-type results and open problems.

After describing the fundamental problem, Michael gives a compelling visual proof, due to Steve Fisk, that a closed polygonal art gallery with n walls requires at most n/3 guards. Next he considers the more subtle question of right-angled galleries (i.e., where adjacent walls meet orthogonally), for which the corresponding maximum number of guards is n/4. Then onto fortresses, where the guards are posted outside the perimeter of the polygonal structure and the object is to protect the exterior and a clever inversion argument is needed. Ultimately the author explores the three-dimensional analogue of Klee?s original problem and produces a surprising example of a polyhedron?the ?octoplex? of the title?where, unlike the two-dimensional situation, posting a guard at every vertex does not guarantee that the entire interior is protected. Indeed, relatively little is known in the three-dimensional case and Michael concludes the article with several open questions about three-dimensional fortresses suitable for student exploration.

Michael has provided a wonderful overview of some recent work in combinatorial geometry that is accessible to a wide audience. The article is highly engaging, exciting, and most deserving of the PÃ³lya Award.

Biographical Note

T. S. Michael received his B.S. from Caltech in 1983 and his Ph.D. from Wisconsin in 1988 under the direction of Richard Brualdi. His research focuses on combinatorics, especially combinatorial problems involving matrices or geometry. He has been on the mathematics faculty at the U.S. Naval Academy since 1990, where he coached the Putnam team for ten years and was the founding coach of the Naval Academy triathlon team. His book, How to Guard an Art Gallery and Other Discrete Mathematical Adventures, was published in 2009.

Think back to your childhood and the game of Chutes and Ladders, in which you used a spinner to try to move your token to exactly square 100 of a 10 Ã? 10 board before your opponents did. Along the way there were ladders to help you jump ahead, but also chutes to send you back. If you were unlucky, the game could be agonizingly slow. Might there be a better way?

After providing a brief review of the game, Cheteyan, Hengeveld, and Jones extend a Markov chain model that uses the ?official? 1-to-6 spinner to one that uses an arbitrary spinner labeled 1 to n in order to understand the relation between spinner range and the expected number of turns for the game. They discover that a spinner with range 1 to 15 will provide the impatient player with the shortest game on average (with an expected length of 25.81 turns). Readers are invited to consider additional variations on their own, and to model other childhood board games, aided by modifiable Maple code provided by the authors.

Because the Markov chain for Chutes and Ladders has 101 states, the authors cleverly explicate their analysis with a simplified 10-state version of the game, making the article exceptionally clear and very enjoyable to read. Cheteyan, Hengeveld, and Jones are to be congratulated for their innovative and enticing introduction to a classical mathematical topic that all undergraduates should see.

Biographical Notes

Leslie A. Cheteyan received her B.S. in mathematics from Montclair State University in 2008, followed by her M.S. in 2011. From an early age she has had a love for math and its implications. Besides math, Leslie enjoys playing all types of sports, though basketball is her favorite. Her competitive nature helps to fuel her motivation in different areas of mathematics. She now works at Memorial Sloan-Kettering Cancer Center in New York as a research assistant.

Stewart Hengeveld received his B.S. in mathematics from Montclair State University in 2008 and his M.S. in 2012. He has enjoyed spending the last seven years as a mathematics and physics tutor at Bergen Community College, and the last four years as an adjunct professor there. During his time at Montclair, he worked as a fellow in the NSF sponsored GK-12 program. In his spare time, he enjoys astrophotography and playing games of all sorts. He now works for Blue Cross Blue Shield in New Jersey. Per aspera ad astra.

Michael A. Jones just completed his 4th year as an Associate Editor at MathematicalReviews in Ann Arbor. Previously he held faculty positions at the U.S. Military Academy at West Point, Loyola University (Chicago), and Montclair State University. He is a graduate of Santa Clara University (B.S., 1989) and Northwestern University (Ph.D. in game theory, 1994). He likes the challenge of examining everyday observations through a mathematical lens and, when appropriate, writing about them. After eight years of living next to a piano teacher in New Jersey, he finally started taking lessons last year.